Soil Steel Composite Bridges
A comparison between the Pettersson-Sundquist design method and the Klöppel & Glock design method including finite element modelling
AMER H. H. WADI
Master of Science Thesis
Stockholm, Sweden 2012
Soil Steel Composite Bridges
A comparison between the Pettersson-Sundquist design method and the Klöppel & Glock design method including finite element modelling
Amer H. H. Wadi
June 2012
TRITA-BKN. Master Thesis 354, 2012 ISSN 1103-4297
ISRN KTH/BKN/EX-354-SE
©Amer H. H. Wadi, 2012
Royal Institute of Technology (KTH)
Department of Civil and Architectural Engineering Division of Structural Engineering and Bridges Stockholm, Sweden, 2012
Cover photo source: SuperCor catalogue www.atlanticcivil.com.au
Preface
In the name of Allah, the Most Gracious and the Most Merciful.
First, I thank Allah for endowing me with health, strength, patience, and knowledge to complete this work.
I would like to express my sincere gratitude to my supervisor, Professor Lars Pettersson of Skanska Sverige AB for given me the opportunity to do this thesis under his kind supervision that with his constant support, guidance, passion and advice, this research study is completed.
I would like to address my genuine appreciation to the support team in ViaCon AB for their support and feedback.
I would like also to thank Professor Raid Karoumi of the Royal Institute of Technology for his guidance and suggestions improving the report.
In addition, thanks to John Leander and Mohammed Safi, PhD students at the Royal Institute of Technology for the interesting discussions, their support and feedback.
Sincere gratitude to my dear friends and classmates for their backing and company in the pursuit of this academic achievement.
Last but not lease, I would like to express my deepest appreciation and love to my beloved parents, brothers and sisters, who have always supported me through times. It is their unconditional love and support that have always encouraged me achieving my goals.
Stockholm, June 2012 Amer H. H. Wadi
Abstract
The need of exploring efficient solutions to today’s engineering problems is becoming essential in the current market development. Soil Steel composite bridges (SSCB) are considered well competitive in terms of their feasibility and constructability. The primary objective of this study is to provide a comprehensive comparison study for two known design methods of SSCB, which are the Pettersson-Sundquist design method (developed in Sweden) and the Klöppel & Glock design method (developed in Germany). Moreover, in the goal of having better behaviour understanding for SSCBs, the study also include finite element modelling (FEM) using PLAXIS 2D of three case studies and compare model results with field measurements.
The design comparison deals with the design concepts, assumptions and limitations for both design methods, where full design procedures are implemented and compared for a defined case study.
The results of the FEM analysis show rational outcome to the field measurements for structural response during backfilling and close results for ordinary loading as well.
While the design comparison shows how the different approach in both design methods in limitations and design assumptions has important impact on the results, where soil failure in the Klöppel & Glock design method can be controlling the design for low heights of cover, whereas formation of crown plastic hinge is more controlling in the Pettersson-Sundquist design method. However, and in general, the Pettersson- Sundquist design method require more steel in low heights of cover while it is less demanding for higher soil covers compared to the Klöppel & Glock design method.
Keywords: Soil-steel, Flexible bridge, Culvert, PLAXIS, Pettersson-Sundquist design method, Swedish design method, Klöppel and Glock.
Contents
Preface ... i
Abstract ... iii
1 Introduction ... 1
1.1 General ... 1
1.2 Background ... 1
1.2.1 The feasibility of SSCB ... 3
1.3 Aims and goals ... 5
1.4 SSCB profiles and types ... 5
1.5 Structure of the thesis ... 8
2 Pettersson-Sundquist design method ... 9
2.1 General ... 9
2.2 Pettersson-Sundquist design method development ... 9
2.3 Principles for design input and analysis ... 10
2.3.1 Backfill soil material ... 10
2.3.2 Loads and loads distribution ... 11
2.4 Limitations and assumptions ... 13
2.5 Design requirements and verifications ... 15
2.5.1 Yielding of pipe wall (SLS) ... 15
2.5.2 Plastic hinge in upper pipe part (ULS) ... 16
2.5.3 Buckling capacity in lower pipe part (ULS) ... 16
2.5.4 Bolted capacity check (ULS) ... 16
2.5.5 Fatigue ... 17
2.6 Case study calculations ... 18
3 Klöppel and Glock design method ... 19
3.1 General ... 19
3.2.2 Load system classification ... 21
3.2.3 Loads and loads distribution ... 22
3.2.4 Limitations and assumptions ... 23
3.3 Design requirements and verifications ... 26
3.3.1 Pipe stress during backfilling ... 26
3.3.2 Snap-through load capacity at crown ... 28
3.3.3 Bolt connection capacity... 29
3.3.4 Bearing capacity of soil for small cover heights ... 30
3.3.5 Fatigue ... 31
3.3.6 Bevel cut ends ... 31
3.4 Case study calculations ... 32
4 Finite element modelling (FEM) ... 33
4.1 General ... 33
4.2 Basics & Potentials “PLAXIS 2D” ... 33
4.2.1 The model ... 33
4.2.2 Elements... 34
4.2.3 Soil material models... 34
4.2.4 Soil steel material interaction ... 37
4.2.5 Steel plate profile modelling ... 37
4.2.6 Construction stages ... 38
4.2.7 Compaction effects ... 38
4.2.8 Boundary conditions ... 39
4.3 Skivarpsån case modelling ... 39
4.3.1 Material properties and geometry ... 40
4.4 Enköping case modelling ... 42
1.1.1 Material properties and geometry ... 42
4.5 Klöppel & Glock test case modelling ... 44
4.5.1 Material properties and geometry ... 45
5 Results and discussions ... 48
5.1 Finite element modelling outcome ... 48
5.1.1 Skivarpsån case results ... 48
5.1.2 Enköping case results... 55
5.1.3 Klöppel and Glock test case results ... 58
5.2 Design methods comparison ... 60
5.2.2 Structure-soil stiffness comparison ... 62
5.2.3 Loads and load distribution comparison ... 64
5.2.4 Sectional forces comparison ... 65
5.2.5 Overall design comparison ... 66
6 Conclusions ... 72
6.1 General ... 72
6.2 Finite element modelling ... 72
6.3 Design methods ... 73
6.4 Further research ... 74
Bibliography ... 75
A. Appendix ... 79
A.1 Charts from K&G ... 79
A.2 Additional FEM results ... 82
B. Appendix ... 86
B.1 Pettersson-Sundquist Design method (MathCAD sample calculations) ... 86
B.2 Klöppel & Glock design method (MathCAD sample calculations ... 105
List of Figures
Figure 1.1: K&G test to failure, 6 m span. Germany 1963. Source: (Klöppel and Glock, 1970) . 2 Figure 1.2: Enköping pipe arch culvert test in Sweden 1987-1990, 6.1 m span Source:
(Pettersson, 2007) ... 3
Figure 1.3: Classification of Trafikverket bridge stock considering the bridge type and construction material. Source: (Safi, 2012, p. 7) ... 4
Figure 1.4: Bridge types for the different span length in Sweden. Source: (Safi, 2012, p. 10) ... 4
Figure 1.5: Different shapes of steel profiles. Source: (Beben, 2009) ... 5
Figure 1.6: Common types of corrugations. Source: (Pettersson and Sundquist, 2010) ... 6
Figure 1.7: Ultra-Cor vs. Super-Cor vs. 152 x 51 mm Corrugation Profile. Source: (Williams et al., 2012) ... 6
Figure 1.8: Mallet River Bridge, 18 m span, Rothsay Ontario, Canada. Source: www.armtec.com ... 7
Figure 1.9: A 52 ft span (15.8 m) bridge, New Hampshire, USA. Source: www.contech-cpi.com 7 Figure 1.10: Whitehorse Creek Arch: 24.0 m span, 12.0 m rise, 30.48 m length, 93S SuperCor Arch, 4.0 m height of cover, Alberta, Canada. Source: www.ail.ca ... 7
Figure 2.1: Equivalent line load and crown pressure from LM1 EN 1991-2, regenerated by discrete points after Pettersson and Sundquist (2010) figure B4.9 ... 12
Figure 2.2: Fatigue LM3 EN 1991-2. Source: EN 1991-2 ... 12
Figure 2.3: Equivalent line load and crown pressure for LM3 EN 1991-2 (axle loads are considered to be concentrated loads) ... 13
Figure 2.4: The fatigue test specimen and the test jig in the KTH Structural Engineering and Bridges laboratory. Source: (Pettersson et at., 2012) ... 17
Figure 3.1: Constrained modulus to elastic modulus ratio ... 20
Figure 3.2: Load system classification. Source: (Klöppel and Glock, 1970) ... 21
Figure 3.3: Live load distribution according to ARS 20/97. Source: ARS 20/97... 22
Figure 3.4: Vertical stress caused by a flexible strip load. Source: (Das, 2006) ... 22
Figure 3.5: Vertical stress-load ratio comparison ... 23
Figure 3.6: Soil pressure and soil heights during backfilling ... 27
Figure 3.7: Active radial pressure for ellipse shape. Source: (Klöppel and Glock, 1970, p. 38) . 28 Figure 3.8: The three design assumptions. Source: (Klöppel and Glock, 1970) ... 29
Figure 3.9: Bolt connection load capacity according to ViaCon design template ... 30
Figure 3.10: Soil failure curve. Source: (Klöppel and Glock, 1970) ... 31
Figure 4.1: Example of a plane strain (left) and axisymmetric problem (right). Source: PLAXIS manual ... 34
Figure 4.2: Position of nodes and stress points in soil elements. Source: PLAXIS manual ... 34
Figure 4.3: Basic idea of an elastic perfectly plastic model. Source: PLAXIS manual ... 35
Figure 4.4: Stress circle at yield, Mohr-Coulombs envelop. Source: PLAXIS manual ... 35
Figure 4.5: The Mohr-Coulomb yield surface in principal stress space c =zero. Source: PLAXIS manual ... 36 Figure 4.6: Hyperbolic stress- strain relation in primary loading for a standard drained triaxial
Figure 4.7: Inflexible corner point (left) and flexible corner point (right) stress improvement.
Source: PLAXIS 2D manual ... 37
Figure 4.8: Position of nodes and stress points in plate element. Source: PLAXIS 2D manual 37 Figure 4.9: Vibratory plate compactors sample data sheet. Source: www.sakainet.co.jp ... 39
Figure 4.10: Skivarpsån geometry and the quarter point location (not to scale). The figure shows the inner measures. Source: (Flener, 2004) ... 41
Figure 4.11: Skivarpsån PLAXIS 2D model, mesh size is medium. ... 42
Figure 4.12: Enköping culvert geometry. Radius measures from wall centre. Source: (Pettersson, 2007) ... 43
Figure 4.13: Enköping PLAXIS 2D model, mesh and connectivity plot, mesh size is medium. 44 Figure 4.14: K&G pipe arch case geometry and live loading scheme. Source: (Klöppel and Glock, 1970)... 46
Figure 4.15: K&G test case PLAXIS 2D, mesh and connectivity plot, mesh size is medium. ... 47
Figure 5.1: Skivarpsån, crown deformation results during backfilling ... 49
Figure 5.2: Skivarpsån, PLAXIS 2D vertical deformation shape when backfill at crown. ... 49
Figure 5.3: Skivarpsån Axial force during backfilling, compaction effect included. ... 51
Figure 5.4: Skivarpsån, PLAXIS 2D Axial force diagram when backfilling is complete ... 51
Figure 5.5: Skivarpsån, Moment during backfilling, compaction effect included ... 52
Figure 5.6: Skivarpsån, PLAXIS 2D Moment diagram when backfilling at crown. ... 52
Figure 5.7: Static test IDs & locomotive axle locations. Source: (Flener, 2004) ... 53
Figure 5.8: Locomotive load distribution and location ... 53
Figure 5.9: Skivarpsån, crown deformation under loading ... 54
Figure 5.10: Skivarpsån, crown axial forces under loading ... 54
Figure 5.11: Skivarpsån, crown moment under loading ... 55
Figure 5.12: Enköping, crown deformation results during backfilling... 55
Figure 5.13: Enköping, PLAXIS 2D vertical deformation shape when backfill at crown... 56
Figure 5.14: Enköping, Axial force during backfilling, compaction effect included ... 56
Figure 5.15: Enköping, PLAXIS 2D Axial force diagram when backfilling is complete ... 57
Figure 5.16: Enköping, Moment during backfilling, compaction effect included ... 57
Figure 5.17: Enköping, Moment diagram when backfill at crown, compaction effect included . 57 Figure 5.18: K&G test case, Maximum crown vertical deflection ... 59
Figure 5.19: K&G test case, Maximum crown moment during backfilling ... 59
Figure 5.20: K&G live load test model, schematic ... 60
Figure 5.21: Comparison of characteristic range of soil tangent modulus. ... 63
Figure 5.22: Stiffness ratio (K&G) and Stiffness number (SDM) for the pipe arch study case . 64 Figure 5.23: Arching effect factor in SDM ... 64
Figure 5.24: Characteristic live load vertical crown pressure comparison ... 65
Figure 5.25: Characteristic live total normal force comparison ... 66
Figure 5.26: Design comparison chart for base coarse material backfill ... 68
Figure 5.27: Design comparison chart for crushed rock material backfill... 68
Figure 5.28: Design comparison chart for sub-base material backfill ... 69
Figure 5.29: Soil capacity against soil failure for different friction angles (regenerated for the pipe arch study case only) ... 70
Figure 5.30: Minimum required plate thickness in K&G for different values of friction angle .. 70
Figure 5.31: Design comparison chart for base coarse material backfill, Steel grade S275 ... 71
Figure A.1: Critical crown pressure for pipe arch culvert. Source: (Klöppel and Glock, 1970) . 79 Figure A.2: Critical normal force for pipe arch culvert. Source: (Klöppel and Glock, 1970) ... 80 Figure A.3: Sample load calculations (bridge load class 60/30) according to K&G. Source:
Figure A.6: Skivarpsån, maximum Crown deformation/moment for different soil elastic
modulus during backfilling ... 82 Figure A.7: Skivarpsån, maximum Crown deformation/moment for different soil friction angles during backfilling ... 83 Figure A.8: Skivarpsån, maximum Crown deformation/moment for different soil cohesion during backfilling ... 83 Figure A.9: Enköping, maximum Crown deformation/moment for different soil elastic modulus during backfilling ... 84 Figure A.10: Enköping, maximum Crown deformation/moment for different soil friction angles during backfilling ... 84 Figure A.11: Enköping, maximum Crown deformation/moment for different soil cohesion during backfilling ... 85
List of Tables
Table 2.1: Development of Swedish Design Method (SDM). Source: Pettersson, L. ... 10
Table 2.2: Main limitations and assumptions in SDM ... 15
Table 3.1: K&G design limitations and assumptions ... 26
Table 4.1: Skivarpsån bridge, material properties and geometry. Source: (Flener, 2003) ... 40
Table 4.2: Skivarpsån PLAXIS 2D material input ... 41
Table 4.3: Enköping, material properties and geometry ... 43
Table 4.4: Enköping, PLAXIS 2D material input ... 44
Table 4.5: K&G test case, material properties and geometry ... 45
Table 4.6: K&G test case, PLAXIS 2D material input ... 46
Table 5.1: Skivarpsån, maximum crown vertical displacement values during backfilling. ... 50
Table 5.2: Skivarpsån, maximum sectional forces values during backfilling. ... 52
Table 5.3: Enköping, maximum crown vertical displacement values during backfilling. ... 56
Table 5.4: Enköping, maximum sectional forces values during backfilling ... 58
Table 5.5: Design limitations comparison ... 62
Notations
Common notations
Abolt Cross section area of bolt [mm2]
As Cross section area of steel profile [mm2/mm]
csp Wave length [mm]
dbolt Diameter of bolt [mm]
Es, Est Nominal elastic steel modulus [GPa]
esp Calculation parameter [-]
fub Bolt ultimate steel capacity according to EN 1993-1-8 [MPa]
fuk Characteristic ultimate strength of steel material [MPa]
fyb Bolt yield steel capacity according to EN 1993-1-8 [MPa]
fyd Design yield strength of the steel material [MPa]
fyk Characteristic yield strength of steel material [MPa]
hcorr Steel profile height [mm]
Is Moment of inertia for steel profile [mm4/mm]
K&G Klöppel and Glock design method [-]
mtt Tangential length of steel profile [mm]
nbolt Number of bolts per meter of culvert width [No’s/m]
r Radius of curvature of steel profile to section centre [mm]
Rsp Radius of curvature of steel profile [mm]
SDM Swedish design method developed by Pettersson and Sundquist
[-]
SDM manual Swedish design method manual developed by Pettersson and Sundquist, 4th edition, 2010.
[-]
SSCB Soil steel composite bridge [-]
ts Steel profile thickness [mm]
Ws Elastic section modulus of steel profile [mm3/mm]
Zs Plastic section modulus of steel profile [mm3/mm]
α Angle for cross section definition [degree]
Pettersson-Sundquist design method notations
abolt Bolts distance to other bolts parallel line (centre to centre)
[mm]
C.mo, C.my Uniform moment factors according to EN 1993-1-1 [-]
C.yy Correction factor according to EN 1993-1-1 [-]
cu Soil uniformity coefficient [-]
D Diameter or span of the culvert [m]
d10 Aggregate size at 10% passing [mm]
d50 Aggregate size at 50% passing [mm]
d60 Aggregate size at 60% passing [mm]
e1 Soil void ratio [-]
Esd Design soil tangent modulus [MPa]
Esk Characteristic soil tangent modulus [MPa]
f.b.Rd Bearing bolt design capacity according to EN 1993-1-8 [kN]
F.st, F.t.Ed Tension in the bolt due to external moment [kN]
F.sv, F.v.Ed Shear in the bolt due to external normal force [kN]
f.t.Rd Tension bolt design capacity according to EN 1993-1-8 [kN]
F.v.Rd Shear bolt design capacity according to EN 1993-1-8 [kN]
f1,f2,f3,f4 Functions used as a mean of simplification [-]
H Vertical distance between crown and spring opening [m]
h.c.red Reduced height of soil cover [m]
h.f Bolt connection vertical distance [m]
hc Soil cover [m]
k.yy Interaction factor according to EN 1993-1-1 [-]
kv Calculation parameters [-]
m1 Soil modulus number [-]
Md Design value for moment [kNm/m]
Ms.cover Moment due to soil cover [kNm/m]
Ms.surr Moment due to surrounding soil [kNm/m]
Msoil Moment due to soil [kNm/m]
Mt Moment due to live load [kNm/m]
Mt.fatigue Moment due to fatigue load [kNm/m]
Mu Plastic moment capacity of steel section [kNm/m]
N.Ed, My.Ed Design values for axial force and bending moment according to EN 1993-1-1
[-]
N.obs Number of heavy vehicles per year per slow lane as per EN 1993-2
[-]
Ncover Normal force due to soil cover [kN/m]
Ncr, N.cr.y Critical buckling load [kN/m]
N Buckling load per unit length of the pipe culvert [kN/m]
Nsurr Normal force due to surrounding soil [kN/m]
Nt Normal force due to live load [KN/m]
Nt.fatigue Normal force due to fatigue load [kN/m]
Ptraffic Equivalent line surface load [kN/m]
Ptraffic.fattigue Equivalent line surface load for fatigue load [kN/m]
q Distributed pressure from traffic [kN/m2]
Q.0, N.0 Traffic values as per EN 1993-2 [-]
Q.m1 Average gross weight of lorries in slow lane as per EN 1993-2
[kN]
R Top radius of the culvert [m]
rd Dynamic factor [-]
RP Degree of compaction according to standard Proctor %
Rs Side radius [m]
Rt Top radius [m]
Sar Arching effect factor [-]
SLS Serviceability limit state
Sv1 Calculation parameter [-]
t.Ld Design life [years]
ULS Ultimate limit state
w.y Plastic to elastic section modulus ratio according to EN 1993-1-1
[-]
α.bu Imperfection factor according to EN 1993-1-1 [-]
ϐ Soil stress exponent [-]
γ.Ff, γ.Mf Partial factors for fatigue according to EN 1993-2 and EN 1993-1-9
[-]
γ.m Soil material safety factor [-]
γ.m.steel Steel material safety factor according to BSK 07 [-]
γ.M1 Steel material safety factor according to Eurocode [-]
γ.n Safety class actor according to BSK 07 [-]
δ.max Maximum theoretical deflection according to EN 1993-1-1 [mm]
ΔF.d.v Shear force per bolt for fatigue [kN]
ΔF.E.2.b Tension force per bolt for fatigue [kN]
ΔM.d.f Design value for fatigue moment [kNm/m]
ΔN.d.f Design value for fatigue normal force [kN/m]
Δσ.c Detail category for tension as per 1993-1-9 [MPa]
Δσ.c.p Plate section category as per EN 1993-1-9 [MPa]
Δσ.E.2.b Tension stress per bolt for fatigue [MPa]
Δσ.E.2.p Fatigue design value for fatigue modified stress as per EN [MPa]
1993-2
Δτ.c Detail category for shear as per EN 1993-1-9 [MPa]
Δτ.E.2.b Shear stress per bolt for fatigue [MPa]
η.m Calculation parameter to check handling stiffness [m/kN]
Κ,ξ,η,ω Calculation parameters for buckling calculations
λ Damage equivalent factor for fatigue as per EN 1993-2 [-]
λ.buk relative slenderness according to EN 1993-1-1 [-]
λ.f Stiffness number ratio [-]
ν Poisson’s ratio [-]
ρ.1 Weight density of the soil material up to crown height [kN/m3] ρ.2 Mean density of the soil material within region (hc+H) [kN/m3] ρ.cv Mean density of the soil material above crown [kN/m3] ρ.opt Optimum density determined according to standards
proctor
[kN/m3] ρ.s True density of the soil material, common range (25 - 26) [kN/m3]
σ Stress due to soil [MPa]
σ.v Vertical crown pressure [kN/m2]
ϕ Calculation parameter according to EN 1993-1-1 [-]
ϕ.2 Damage equivalent impact factor as per EN 1993-1-9 [-]
ϕ.d Design friction angle of soil [degree]
ϕ.k Characteristic friction angle of soil [degree]
χ.y Reduction factor for flexural buckling according to EN 1993-1-1
[-]
к Calculation parameter [-]
Klöppel and Glock design method notations
c.soil Soil bedding value [kN/m3]
f Shape factor [-]
f.N Calculation parameter for critical normal force [-]
f.p Calculation parameter for critical crown pressure [-]
FOS.vD Factor of safety against snap though/buckling failure [-]
H.u Soil cover [m]
Hs Total height of the pipe [m]
N.BR Bolt capacity of normal force [kN/m]
N.DT Normal force in the pipe [kN/m]
P.1 Soil cover pressure on crown [kN/m2]
2
P.ogr Soil capacity contribution against soil failure [kN/m ] P.otr Total structure capacity against soil failure [kN/m2] P.ov Equivalent germen uniform surface live load [kN/m2]
P.s Total pressure on crown [kN/m2]
P.s.cr Critical crown pressure [kN/m2]
P.SD Design value for crown pressure [kN/m2]
R.bolt Bolt capacity [MPa]
R1 Top radius of the pipe [m]
R2 Corner/side radius of the pipe [m]
R3 Bottom radius of the pipe [m]
Rg Limiting radius during backfilling [m]
S.v Constrained compression soil modulus [MPa]
Ss Span of the culvert [m]
Stiff.ratio Stiffness ratio [-]
v.BR Safety factor against seam failure [-]
α, β, γ Moment and shape factors for soil failure calculations [-]
γ.B Soil density [kN/m3]
ΔP Pipe capacity contribution against soil failure [kN/m2]
ο Sock factor [-]
ρ Friction angle for the soil [degree]
Ψ.B Load distribution factor (angle in radian) [-]
1 Introduction
1.1 General
The world markets demand of an efficient and practical solutions to different engineering problems is becoming intense and supplementary competitive. The market for Soil steel composite bridges (SSCB) has grown rapidly through the years to become one of the feasible and most competitive alternatives in many cases to conventional bridge construction (CSPI, 2007).
Soil steel bridge is a structure compromised of structural steel plate and engineering soil, designed and constructed to induce a beneficial interaction between the two mate- rials serving its ultimate purpose as a bridge or a culvert (CSPI, 2007).
Authorities’ demand of a better and safer investment in these structures has stimulated the engineering research and the industry section into more design and performance investigations. Where after, different design methods have emerged through Europe and American continent as well, and most of these methods still devel- oping from time to time, in order to cover new manufacturing products and new design challenge.
1.2 Background
The research development of buried structures believed to have started as early as 1913 in Iowa State College by Marston, Spangler and others. A significant conclusion in 1923 on the Illinois Central Railroad by the American Railway Engineering Associa- tion (AREA), a conclusion where measurements have shown that flexible corrugated pipes carried only 60% of the 10.7 m column of the fill above, while the remaining percent were carried by the adjacent soil (CSPI, 2007).
In 1960’s, the concept of ring compression theory by Marston-Spangler was introduced (Abdel-Sayed et al., 1994). Further, between 1967 and 1970, extensive research sponsored by American Iron and Steel Institute (AISI), conducted at Utah University, and had a great contribution in this research field (CSPI, 2007).
In Europe, in 1963, investigations were performed under the supervision of K. Klöppel and D. Glock through Technical University Darmstadt (Germany) in cooperation with the Deutsche Bundesbahn and Armco-Thyssen GmbH (Klöppel and Glock, 1970).
They published a report in 1970, covering the load carrying behaviour of flexible earth embedded pipes under soil covers, and they proposed a way of calculating their bearing
Figure 1.1: K&G test to failure, 6 m span. Germany 1963. Source: (Klöppel and Glock, 1970) In 1975, the state of California USA conducted a significant research project called the D.B. culvert with 3050 mm diameter, with a 2.8 mm plate thickness, of which the performance data contributed a lot in development and verifications of new design tools.
The development and the use of finite element analyses in the 1970s and 1980s changed the nature of culvert assessment allowing consideration of the geometrical and material details of the structure condition, as well as the construction process, culvert geometry, and soil and vehicle loads (El-Taher and Moore, 2008). One of the finite elements tools was CANDE. This was a FHWA (Federal Highway Administration) sponsored com- puter program by M. Katona, et al. This introduced the soil culvert interaction (SCI) design method proposed by Duncan, which utilizes design graphs and formulas based on finite element analysis (CSPI, 2007).
The Ontario Highway Bridge Design Code (OHBDC) was first introduced in 1979, which includes a section for the design of SSCB and considers ultimate limit state (ULS) under load combination of dead and live load (Abdel-Sayed et al., 1994).
Other codes and design methods are in use such as American Iron and Steel Institute (AISI), where it includes working stress or service load method (CSPI, 2007). In addition, ASTM International, formerly known as the American Society for Testing and Materials (ASTM) has a procedure for structural design of SSCB under standard designation of ASTM A796 / A796M. A similar method in American Association of State Highway and Transportation Officials (AASHTO) is available, which have a section regulating corrugated steel pipes. Although these methods have some similari- ties, each method has its own limitations and criteria.
SSCB were introduced in Sweden in the mid 1950s, the design was done using simple diagrams and so called standard drawings, and these standards drawings were prepared for spans up to 5 m (Pettersson, 2007).
A design method in Sweden have been developed (often referred as Swedish design
investigations calibrated by several scale tests and finally presented in its 4 edition manual in 2010 by Pettersson and Sundquist. This method includes ULS and SLS calculation check in line with the prevailing European code in steel design (more information about this method is presented later in this report).
Figure 1.2: Enköping pipe arch culvert test in Sweden 1987-1990, 6.1 m span Source:
(Pettersson, 2007)
1.2.1 The feasibility of SSCB
It is estimated that more than 1/3 of the over 600,000 bridges in north America are in need of repair, the majority of the bridges are termed short span, less than 15 m in length and can be replaced by SSCB. The use of large diameter SSCB in real situation as an alternative to bridge replacement has shown in one reported case excellent cost saving of 51% (CSPI, 2007, p. 360).
In Sweden, at the end of 2011, the stock of bridges owned by Trafikverket (The Swedish Transport Administration) is around 20000 bridges (Safi, 2012). Earlier study showed that by 2006 about 2270 culverts out of 2400 culvert owned by Swedish road Administration are made of corrugated steel (Mattsson and Sundquist, 2008).
Obviously, many of the old short span culverts need to rehabilitated or even replaced, and SSCB shows good promises as a good feasible alternative, especially with the current corrosion protection technology in the market.
Moreover, Safi (2012, p. 24) reported that the use of flexible culvert as a replacement for concrete bridge with a 9.3 m span has shown being most cost-effective alternative among other conventional bridges.
Figure 1.3: Classification of Trafikverket bridge stock considering the bridge type and construction material. Source: (Safi, 2012, p. 7)
Figure 1.4: Bridge types for the different span length in Sweden. Source: (Safi, 2012, p. 10)
1 10 100 1000 10000
Concrete Steel Timber Stone Special Material
Number of Bridges
Bridge Construction Material
Slab Bridge Beam Bridge Slab-Frame Bridge Beam-Frame Bridge Culvert Bridge Earth Filled Arch Bridge Open Spandrel Arch Bridge Cable Stayed Bridge Suspension Bridge Other Bridge types
1
10
100
1000
10000
S< 5m 5m ≤ S< 10m 10m ≤ S < 20m 20m ≤ S < 30m 30m ≤ S < 50m 50m ≤ S < 100m 100m ≤ S < 200m 200m ≤ S < 500m 1000m ≤ S
Number of Spans
Span Length (S)
Beam Bridge Beam-Frame Bridge Cable Stayed Bridge
Culvert Bridge Earth Filled Arch Bridge Open Spandrel Arch Bridge
Slab Bridge Slab-Frame Bridge Suspension Bridge
1.3 Aims and goals
In the trend of having a harmonized code for SSCB in Europe, this study aims primary to establish a comprehensive comparison between two dominate design methods in Europe for SSCB which are, the Pettersson-Sundquist design method which often referred to as the Swedish design method (SDM) and the German method known as the Klöppel and Glock design method (K&G).
On the line of serving the purpose of having better understanding of these structures behaviour, a second objective is also involved, which explores finite element modelling (FEM) of SSCBs, where model results are compared to field measurements for different case studies to verify how rational the model is as a tool when comparing to the theoretical results
The study purpose is not to criticize or favour any of the methods; on the contrary, it provides a first scientific comparison based on reasoning between these two methods, including their design philosophy and limitations.
The goal is to highlight the similarities and differences and identify the advantages and drawbacks in both design approaches, in addition to having a better structural percep- tive using the FEM analysis, all in serving the ultimate goals of having a deeper understanding, and providing modern knowledge to the research and the industry sector as well.
1.4 SSCB profiles and types
Soil steel composite bridges can be built in different shapes. Figure 1.5 below shows some of these different shapes, which are categorized between closed and open shapes.
Figure 1.5: Different shapes of steel profiles. Source: (Beben, 2009)
The SDM manual in its 4 edition (Pettersson and Sundquist, 2010), covers seven shape types of these structure, while it worth mentioning that K&G concentrated more on the circular and the pipe arch shapes.
As for the different types of corrugation profiles, there are several types of corrugations that are used commonly in the current market, and their thickness can range from 2 to 7 mm. The following figure shows most used types in the market.
Figure 1.6: Common types of corrugations. Source: (Pettersson and Sundquist, 2010) Higher types of corrugation such as deep corrugated Ultra-Cor structural plate with a corrugation pitch and depth of 500 mm x 237 mm and thickness range from 7 to 12 mm (Williams et al., 2012), where it is being developed to offer solutions where necessity of large spans, lower soil covers and higher live loading are used.
Figure 1.7: Ultra-Cor vs. Super-Cor vs. 152 x 51 mm Corrugation Profile. Source: (Williams et al., 2012)
Larger spans of SSCB are possible now reaching more than 20 m (Pettersson. 2007), which require utilization of the plate profile and soil material, where all being addressed in intensive research and full scale tests.
Figure 1.8: Mallet River Bridge, 18 m span, Rothsay Ontario, Canada. Source:
www.armtec.com
Figure 1.9: A 52 ft span (15.8 m) bridge, New Hampshire, USA. Source: www.contech-cpi.com
1.5 Structure of the thesis
This thesis comprises six chapters and two appendices. The first chapter involves a quick introduction to the market of soil steel composite bridges (SSCB) and highlight some of the advantages of using these structures.
The second and third chapter deals with the Swedish design method and K&G design method respectively, where they underline the main concepts and approach of both design methods, exploring their theory, limitations and calculations as well. In addition, at the end of these chapters, some general words about the adopted case study for the comparison.
The forth chapter is dealing with the finite element modelling (FEM) of SSCB. It includes brief introduction to PLAXIS 2D software, also depicts the input and the methodology of the 2D modelling of mainly three different case studies (Skivarpsån, Enköping and K&G case studies) using PLAXIS 2D.
While the fifth chapter represents the results and discuss the outcome from both FEM analysis in comparison to field measurements and the design comparison results of the two design methods as well.
At the end of thesis, in the sixth chapter, the main conclusions derived from this study are highlighted and summarized.
2 Pettersson-Sundquist design method
2.1 General
The need for a Swedish design method for soil steel composite bridges has emerged to cover the on-going demand of market investment in building these structures. The Swedish road administration - Vägverket (SRA)1 saw a need for more accurate design method, although international design methods were available, but it was unclear how they should be applied to the Swedish conditions.
The challenges to build structures with larger spans and small cover heights were encountered by the limitations and restrictions existing in the international methods.
Moreover, these design methods had some drawbacks like the effect of higher degree of compaction or even effect of soil grading, and in order to have better understanding of the behaviour of these structure, an investigation has been made (beginning of 1980s) for developing a new design method composed together with the international experi- ence and suitable for Swedish conditions.
2.2 Pettersson-Sundquist design method development
The following table summarizes the history line for the main events concerning development of the Pettersson-Sundquist design method (often referred as the Swedish design method) presented till this date in its forth revision by Pettersson & Sundquist (2010) of which the calculation in this report is based on.
Year(s) Event
1983 Full-scale tests in Nyköping, Swedish road administration (SRA) starts research project in cooperation with KTH
1987-1990 Full-scale tests in Enköping (pipe arch bridge).
1998 Load classification method is presented 2000 Swedish design method (SDM) is presented
2002 Box culvert profiles introduced in Sweden and full scale tests are performed
Year(s) Event
2003 Railroad bridge (Skivarpsån) is designed according to SDM including extensive static and dynamic testing were performed.
2004 An extension of SDM to cover Box culverts is proposed.
2004 Swedish bridge code Bro 2004 requires design according to SDM and the same for rail administration through Railroad Bridge code (BV Bro, 8th edition).
2005-2006 Further full scale tests are performed to verify new extensions in SDM 2006 3rd revision of SDM is presented including box culvert extended method 2007 English version of SDM is presented including Eurocode adoption 2010 4th revision of SDM is presented with some refinements and additions.
Table 2.1: Development of Swedish Design Method (SDM). Source: Pettersson, L.
2.3 Principles for design input and analysis
The calculations in SDM are based on three main theories, which were compared and calibrated with full-scale tests and used through the design process; these important points are summarized as follow:
• The soil culvert interaction (SCI) developed by Duncan (1978-1979)
• The buckling calculations presented by Klöppel & Glock (1970)
• Soil modulus for soil frictional material by Andreasson (1973)
• Arching calculations developed by Vaslestad (1990)
This design method has been developed and improved to cope with the prevailing regulations and standards, where it provides ultimate capacity calculations using either the Swedish standards or the European code. By this context, the method is considered as code independent.
2.3.1 Backfill soil material
Generally, in the Swedish design method, there are two ways of calculations for the tangent modulus of structural backfill material and it will depend on the available geotechnical information.
The first method is based on Duncan (1978) which is relatively simple and depends on the degree of compaction of the backfill and this method gives conservative values compared to second and more accurate method that needs more input for the backfill such as sieve analysis, degree of compaction, and stress level in the surrounding fill soil.
By principle, the tangent modulus of soil material depends on the stress state, which changes from point to point. Therefore and for analysis simplification, the SDM adopts
crown and spring line and base the calculation on the stress state evaluated from at- rest pressure just under these quarter points of the culvert.
Obviously, SDM provides the flexibility of changing the soil material properties and more importantly covers the effect of compaction, where it is characterized by interna- tionally recognized method “standard Proctor method”, which is implemented through calculations of evaluating the soil stiffness.
In this report, and in the calculation phase, the second method will be used in assessing the tangent modulus of soil material, which is considered relatively more reliable, and gives results that are less conservative.
2.3.2 Loads and loads distribution
The SDM, concerning loads and load distribution, gives the choice of calculating the effect of any live load whether it is concentrated or distributed.
The Boussinesq’s semi-infinite body method in load distribution is used in a way of converting any live load on the surface by an equivalent line load where they have the same vertical pressure effect on the culvert crown. By doing this, design load models in the prevailing standards (i.e. Eurocode, Swedish code, etc…) all can be investigated and regenerated in curves where soil cover is plotted against the corresponding line traffic load (SDM manual provides similar figures for known load models) and these values are used in the design equations for sectional forces calculations.
2.3.2.1 Live load
In order to make comprehensive calculations, the report will cover the effect of load model 1 in Eurocode EN 1991-2 by studying structural response to such load under different design conditions inclusive of the soil cover effect. The reference manual of SDM provides a figure where equivalent line load is calculated.
The following figure is regenerated and thus vertical crown pressure is back calculated using equation b4.b in the SDM manual, which is going to be used in the design process.
Figure 2.1: Equivalent line load and crown pressure from LM1 EN 1991-2, regenerated by discrete points after Pettersson and Sundquist (2010) figure B4.9
2.3.2.2 Fatigue load
In additions to the main loading scheme above, and since SDM has the advantage of including fatigue load verification, the main loading scheme for such verification is chosen to be fatigue load model 3 in EN 1991-2 and it is a single vehicle model with an axle load of 120 kN
Figure 2.2: Fatigue LM3 EN 1991-2. Source: EN 1991-2
In order to assess the structure capacity for the above fatigue load, and in the same way, the equivalent line load is to be calculated for different soil cover. The following figure represents these calculations where for simplicity, wheel loads are considered as concentrated loads (wheel area is not considered).
0 25 50 75 100 125 150 175 200 225 250
0 50 100 150 200
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 Vertical crown pressure (kN/m2)
"P" surface line traffic load (kN/m)
Soil cover, hc (m)
Equivalent line load and crown pressure for Eurocode load, LM1
P surface line traffic load σv vertical pressure at crown
Figure 2.3: Equivalent line load and crown pressure for LM3 EN 1991-2 (axle loads are considered to be concentrated loads)
It is worth to mention here that the above fatigue load and in comparison to fatigue load in Swedish code Bro 2004 presented in a study by Hirvi (2007), LM3 has slightly less values in the equivalent line load, this is due to higher axle loads for fatigue loading in Bro 2004.
The fatigue calculations in this report require highlighting few design inputs and assumptions, which will be briefly mentioned in section 2.5.5.
2.4 Limitations and assumptions
The Swedish design method has a set of limitation and assumptions with regard to material calculation and geometry of profile shapes. The SDM manual provides details about the limitation and criteria to be aware of during analysis and designing.
In order to concise all the information and to be comparable with other design method (in this case K&G), the following table highlights the major limitations and assump- tions in the SDM with corresponding remarks with regard to each item and criteria.
0 20 40 60 80 100 120 140
0 10 20 30 40 50 60 70 80 90 100
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 Vertical crown pressure (kN/m2)
Equivalent line surface load (kN/m)
Soil Cover, hc (m)
Equivalent line surface load and crown pressure for fatigue loading, LM3
Equivalent line surface load Vertical crown pressure
Item Swedish Design Method
criteria Remarks
General The calculations in this report will adhere to the limitations and criteria provided in SDM, and thus will be comparable with the K&G design method.
Stiffness ratio
3
100 f E Dsd 50000
λ EI
≤ = ≤
Originally, the upper limit was 100 000 (original SCI method) and then reduced, since the bending moment equation be- comes zero at very high stiffness numbers, where this is not ac- ceptable for large span struc- tures.
Steel type corrugation profiles
Most commonly occurring sys- tems using relatively thick cor- rugated steel profiles including 200×55 and 381×140 mm
SDM has the ability of using high profiles by including its sen- sitivity to local buckling.
Geometry limits for pipe arch shapes
R1/R2≤5.5 R3/R1≤10
Since the main case study here is a pipe arch, only information related to this shape is men- tioned, however other limitations for other shapes are stated in the SDM manual.
Minimum
soil cover 0.5 m for road bridge culverts.
1.0 m for railway cases. Higher limit in railway cases to allow for ballast maintenance.
Steel material
Not specified, covers most of common steel material grades with all the applicable safety factors.
In this report, steel grade S355 in accordance with EN 10025 will be used.
Backfilling soil material
Represented by tangent soil modulus, which is dependent on degree of compaction and particle size distribution.
Friction angle is calculated from soil parameters and nor- mally it is the range of 35-45o
In the calculation comparison, tangent soil modulus is calcu- lated using available equations in SDM.
Friction angle value is calculated according to given soil parame- ters.
Live loads
Live loads are taking care of by converting to their equiva- lent line loads using Boussi- nesq’s theory, based on the same pressure effect on culvert crown.
In this report, LM1 in EN 1991-2 will be used; the corresponding equivalent line load is derived and implemented in the calcula- tions.
Fatigue
loads Treated in similar way as live
loads. In this report, LM3 in EN 1991-2
will be used.
Controls during backfilling operations
A serviceability check is made and compared to the selected design allowable yield stress according to steel grade with the applicable safety factors.
In this report calculation com- parison, Steel grade of S355 in accordance with EN 10025 will be used.
Item Swedish Design Method
criteria Remarks
Minimum thickness of steel plate
Not specified, design verifica- tions in SLS and ULS are the main criteria for selecting plate thickness.
Span
Not specified, higher spans are achievable provided adherence to corresponding limitations and design criteria.
Table 2.2: Main limitations and assumptions in SDM
2.5 Design requirements and verifications
The calculations of sectional forces in SDM are mainly based on the original SCI- method with some modifications with respect to load distribution, influence of cover depth and soil modulus calculations, the proposed modifications have been adopted by comparison with results of full-scale test as reported in SDM manual.
It is worthwhile to indicate that the application of SDM require applying partial coeffi- cients to both loads and carrying capacities, these factors are as recommended by the relevant prevailing codes and standards. In this report, verifications will be performed according to applicable Eurocode and thus relevant factors for both SLS and ULS are according to EN 1990.
SDM requires designer to perform certain verifications of the load bearing capacity in both serviceability and ultimate limit states, these verifications are mentioned suffi- ciently in the SDM manual, however, the main points of those are described below and will be used in this report as part of the calculations comparison with the K&G design method.
2.5.1 Yielding of pipe wall (SLS)
The stresses in the pipe wall is calculated according to Navier’s equation and verified with respect to maximum allowed yield stress. The construction nature of those struc- tures would require designer to perform this check to simulate the real case stress de- velopment during construction, meaning that stresses due to soil or combined soil plus traffic are to be verified and fulfilled separately subsequent to the maximum stresses during backfilling and stresses when complete structure is loaded as well. (Refer to cal- culation sample shown in appendix B)
The chosen point to perform such check is normally the crown section with its design normal force and bending moment.
2.5.2 Plastic hinge in upper pipe part (ULS)
The calculation for this criterion is performed in ultimate limit state, where check against development of plastic hinge in the upper part of the pipe is verified. The check is to be performed with interaction formulas according to EN 1993-1-1. More precisely, equation 6.61 will be utilized and simplified due to the assumption that plate is pre- sumed not to deflect laterally. In addition, the section properties of the plates used for the culverts are commonly in cross section class 1 or 2, hence additional moments due to neutral axis shift do not exist.
In this report, the risk of local buckling of profiles is taking care of by reducing the plastic moment capacity of plate whenever required according to formula reported in SDM manual.
The SDM implies a reduction in load bearing capacity due to second order theory moreover require performing interaction check in relation to case where maximum normal force occurs. Calculations regarding this part are more clearly illustrated in the calculation sample presented in appendix B.
2.5.3 Buckling capacity in lower pipe part (ULS)
It is important to check the pipe capacity in terms of global buckling against the pre- vailing normal force in the pipe, where it is assumed that normal force is the same along the profile and equal to the design value calculated due to soil and traffic.
2.5.4 Bolted capacity check (ULS)
Normally the soil steel culvert structure are made from steel plate segments connected with bolts in both directions (commonly 20 mm diameter bolts in 25 mm holes), these bolted connections have to be checked appropriately with respect to the design forces in the pipe.
Since verifications in this report are made according to Eurocode, calculations related to this part are performed mainly in accordance to equations in EN 1993-1-8 with the implementation of partial factors mentioned in the SDM handbook.
The said code dictates to check bolted connections with respect to three conditions and all have to be fulfilled, inclusive of pure shear case, tension case and combined shear- tension as well.
The distances between bolts and bolt to plate edge are chosen and regulated in accor- dance to applicable prevailing standards such as Swedish code BSK 07 and Eurocode EN 1993-1-8.
2.5.5 Fatigue
The current edition of SDM gives the possibility to perform calculations for fatigue capacity under some assumptions. However, and due to the structural nature regarding geometry and load distribution, the fatigue capacity of these structures is being reas- sessed, where recently a deeper study has commenced in KTH aiming to provide a more accurate understanding and design procedures for the fatigue problem verified through testing for this kind of structures.
Figure 2.4: The fatigue test specimen and the test jig in the KTH Structural Engineering and Bridges laboratory. Source: (Pettersson et at., 2012)
Nevertheless, this report will illustrate fatigue verifications according to section 8 in EN 1993-1-9 together with the criteria mentioned in the SDM handbook. The calcula- tions will imply to check fatigue capacity of the structure under fatigue loading de- scribed earlier in section 2.3.2.2 and more precisely will concentrate on fatigue capacity of the plate itself, and by doing so, this will be used as part of the comparison calculation with respect to the K&G design method.
The bolted connection fatigue capacity will not be part of the criteria in the compari- son calculations for the reason explained earlier being under research, however and since it is currently proposed in the SDM handbook, sample calculations are presented in appendix B.
In order to perform the fatigue verification for the plate, some criteria and assumptions have to be highlighted in-line with EN 1993-1-9 and EN 1993-2 summarized as follows:
• Low consequence safe life method is applied
• The moment for the bolted connection will be evaluated at a joint located more than the height of the cover from the riding surface, where SDM suggest possible reduction of the moment at the crown by certain ratio, for calculation purposes a realistic value of 0.85 is assumed and used.
• The design stress is evaluated assuming that minimum moment value is equal to half-maximum moment value as suggested by Pettersson (2007).
• Traffic category number 4 is chosen in relation to EN 1991-2 table 4.5(n).
• The section category for the plate is assumed detail 5 in table 8.1
• Other assumptions are mentioned in the calculations as illustrated in appendix B.
2.6 Case study calculations
The main part of SDM calculations is performed on single case study presenting Enköping pipe arch culvert geometry with the range of plate profile 200×55 mm, the calculations will cover the effect of different heights of soil cover and the corresponding design thickness for the said profile.
The use of Eurocode in the capacity verifications will be employed according to SDM manual. Moreover, since the design verify the ULS and the SLS, all applicable load and material factors will be utilized according to design manual and applicable Eurocode and will be stated and indicated in the calculation sample presented in appendix B in this report.
Moreover, and for the sake of comparison, typical soil material mentioned in the SDM manual table B2.1 will be used in the main calculation body for assessing the design method in comparison with the K&G method.
The main results of the SDM calculations will be presented, compared and discussed in chapter 5.
3 Klöppel and Glock design method
3.1 General
In 1970, K. Klöppel and D. Glock through Technical University Darmstadt (Germany) and in cooperation with the Deutsche Bundesbahn, published a report covering the load carrying behaviour of flexible earth embedded pipes under different soil covers and they proposed a way of calculating their bearing capacities for design purposes.
The investigation was performed by Armco-Thyssen GmbH Company, which was a foundation in 1957 between Armco Inc., Ohio (USA) and August Thyssen Hütte AG, Duisburg (Germany), later on, this venture got segregated and in 1987 new company has established under new name “Hamco Dinslaken Bausysteme GmbH”2.
Nevertheless, and in 1963 a large-scale test (Klöppel and Glock, 1970) supported by Armco-Thyssen, underlined the method and gave the fact of the high carrying capacity for static loading, which laid down to adoption in many European countries standards for corrugated steel pipe.
In order to make the investigation in the report up to date, a connection has been made with one of the current leading companies in this sector. ViaCon Austria have provided their advice and latest information with regard to the current market condi- tions concerning the implementation of K&G design method, knowing that ViaCon Austria base their design on the K&G method together in association with applicable regulations and standards. One of the main regulations governing the design and was based on K&G method is ARS 20/97 and its addition ARS 12/98 which are guidelines concerning the conditions for the use of corrugated steel pipes dimensioning instructions.
The design calculations in this report are based on the K&G design method with the additions indicated in the German specifications said earlier ARS 20/97 & ARS 12/98.
Moreover, in order to be consistent and in harmony with the current practice in the market, the calculations are checked and cross-referenced with an excel design template developed by ViaCon Austria, which is based on K&G design method in conjunction with the German specification mentioned earlier.
3.2 Principles for design input and analysis
This section will briefly describe the main concepts, assumptions and limitations of which K&G design method is based on; in addition, of what was reported in the original report, this section will also highlight few topics related to the current practice of this method.
3.2.1 Backfill soil material
This method uses the constrained soil modulus Sv as stiffness representation for the backfilling soil material, which can be defined as the deformation modulus of the compression test with obstructed lateral expansion. This modulus is little higher than the tangent modulus of soil Es.
According to K&G report, the elastic modulus Es for soil material such as gravel and sand is in the range as shown in equation (3.1) below
(0.64 0.775)
s v
E = → S (3.1)
On the other hand, Pettersson (2007) reported similar values as function of soil Poisson’s ratio as shown in equation (3.2) below, where ν is the soil Poisson’s ratio, which can be evaluated for frictional material can be evaluated as function of friction angle ϕ as shown in equation (3.3)
1 2 2
s 1 v
ν ν
E S
ν
− −
= − (3.2)
1 sin 2 sin ν φ
φ
= −
− (3.3)
Figure 3.1 shows the close range as K&G for the ratio variance for different friction angles according to equations above.
0.4 0.6 0.8 1.0 1.2 1.4 1.6
29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 Ratio (Sv/Esoil)
Soil friction angel, φ (degree)
Constrained modulus (Sv) to Elastic modulus (Esoil)
According to ARS 20/97 and in general for design purposes, soil backfilling material has to be suitable for backfilling purposes and fall either in coarse grained soil types or mixed grained soil types (soil groups similar to the unified classification system for soil material) with maximum aggregate size of 63 mm and uniformity coefficient Cu more than 3.0. while, according to same specification Sv for soil can be selected as low as 20 MPa with friction angle ϕ=30o, and this Sv value can be chosen higher to be 40 MPa in cases where plate test (DIN 18134) shows initial modulus Ev more than 30 MPa. Of course, designer has to be aware that this higher value is to be validated by the said plate test during construction.
One of the main principles in the K&G method is that the soil material around the pipe is represented by soil bedding stiffness (c), and it is stated that for gravel and sand material this variable can be calculated using equation (3.4) below.
0.5Sv
c = R (3.4)
3.2.2 Load system classification
In general, the level of live loading compared to the soil cover classifies the structural system into two main categories.
The first category is low fill soil cover, and the criteria for this is that soil pressure from dead weight of soil at crown (γ Hu) is less than the equivalent surface uniform live load Pov. In this case the crown pressure (from soil dead weight plus pressure from live load) at the crown is increased by 10% mainly to allow for load concentration and load dynamic effect, and the crown pressure for this case at pipe upper part will be distrib- uted over a smaller angle denoted as ψB=1.57 radians.
In cases, where the dead soil pressure is more than equivalent surface uniform live load Pov, this case is referred as high fill soil cover. In this situation, the loading on the upper part of pipe is distributed over a wider angle and it is denoted as ψB=2.36 radians, and in this case the crown pressure is calculated simply from the summation of soil dead weight and crown pressure from surface live load.
